I need to prevent mobs from getting into my shelter, and I have an exposed lower area with a ladder leading up to the shelter proper. If a zombie or something spawns down there at night, can it reach me in my shelter?

Minecraft mobs are not technically smart enough to climb ladders. However, they can move forward, and moving forward is all they have to do to use ladders/vines.

As was mentioned above, this means that they cannot path to you if they have to go out of their way to use a ladder. But, as was likely in John the Green's case, if you run away up a wall and then back away from the ladder, the mob will push against the wall to get you and inadvertently use the ladder. A simple sidestep would have pulled the mob off of the ladder before he reached the top.

Try using a trapdoor at the top of an interior ladder. Fences and Fence gates are also great barriers at the bottom of the ladder to keep the bad guys out without obstructing your vision.